Post on 13-Mar-2020
Bayesianmodel choice
J. O’Quigley
OverviewContinualreassessmentmethod
Subjectheterogeneity
2 group model
SimulationsImpact of prior
Comparisonwith optimaldesign
CombinationtherapiesPartial ordering ofdoses
Models
Simulations
Using information ongrades
Bayesian model choice for generalizedCRM models
John O’Quigley
MSKCC, New York 2009 1 / 25
Bayesianmodel choice
J. O’Quigley
OverviewContinualreassessmentmethod
Subjectheterogeneity
2 group model
SimulationsImpact of prior
Comparisonwith optimaldesign
CombinationtherapiesPartial ordering ofdoses
Models
Simulations
Using information ongrades
Bayesian model choice/model selection/modeladaptive/assessment/diagnostic
Refs: Kass and Raftery, Gelfand, Gelfand and Ghosh,others
Model averaging (Yin and Yuan 2009)
Patient heterogeneity
Bridging studies
Multi-drug problem, partial ordering
Graded toxicities
MSKCC, New York 2009 2 / 25
Bayesianmodel choice
J. O’Quigley
OverviewContinualreassessmentmethod
Subjectheterogeneity
2 group model
SimulationsImpact of prior
Comparisonwith optimaldesign
CombinationtherapiesPartial ordering ofdoses
Models
Simulations
Using information ongrades
Continual Reassessment Method
Typically a unique working model is assumed for theunderlying true dose-toxicity relasionship before thestart of the trial:
Pr(Yj = 1|Xj = xj) = ψ(di ,a) = αai .
Model parameter(s) a sequentially updated afterobservations on each subject or group of subjects.
Subject allocation uses current running esimate ofMTD.
MSKCC, New York 2009 3 / 25
Bayesianmodel choice
J. O’Quigley
OverviewContinualreassessmentmethod
Subjectheterogeneity
2 group model
SimulationsImpact of prior
Comparisonwith optimaldesign
CombinationtherapiesPartial ordering ofdoses
Models
Simulations
Using information ongrades
Simple patient heterogeneity: 2 groups
A dose finding study for breast cancer treatment at UVA CC.
Goal: to determine the MTD of dasatinib.
Two groups of patients
Dasatinib plus capecitabine for paclitaxel-refractory metastatic breastcancer patients (G1)Dasatinib plus fulvestrant for hormone-sensitive, progressivemetastatic breast cancer patients (G2)
G1 patients are more sensitive to dasatinib and may have a higherprobability of toxicity.
G2 patients may have higher MTD than G1 patients.
It is believed the difference in MTD would not exceed 2 doses.
MSKCC, New York 2009 4 / 25
Bayesianmodel choice
J. O’Quigley
OverviewContinualreassessmentmethod
Subjectheterogeneity
2 group model
SimulationsImpact of prior
Comparisonwith optimaldesign
CombinationtherapiesPartial ordering ofdoses
Models
Simulations
Using information ongrades
A 2-group CRM shift model
Power model for the one-sample CRM model:
ψ(xj , a) = αai (1)
Power model for the two-sample CRM shift model:
ψ(xj , a) = αaφ(i), i = 1, ..., k , φ(i) = i + zj∆(i) (2)
where:
∆(i) denotes the doses shifted from dose i and it takes integer valuebetween −k + 1 and k − 1.zj is an indicator for groups and takes value of 0 or 1.0 < αm < αn < 1 for m < n.0 < a <∞.
If the order of the difference is known, the sign of ∆(i) is known.If zj = 0 group is more sensitive to the treatment, ∆(i) ≤ 0.If zj = 0 group is less sensitive to the treatment, ∆(i) ≥ 0.We may be able to limit ∆(i) to very few values.
MSKCC, New York 2009 5 / 25
Bayesianmodel choice
J. O’Quigley
OverviewContinualreassessmentmethod
Subjectheterogeneity
2 group model
SimulationsImpact of prior
Comparisonwith optimaldesign
CombinationtherapiesPartial ordering ofdoses
Models
Simulations
Using information ongrades
Particular situations
Bridging studies
Several prognostic groups: ∆1,∆2,∆3, ...,
Several groups, some having known orderings
Continous prognostic variable broken into classes
Different treatment schedules
Schedules × prognostic groups
MSKCC, New York 2009 6 / 25
Bayesianmodel choice
J. O’Quigley
OverviewContinualreassessmentmethod
Subjectheterogeneity
2 group model
SimulationsImpact of prior
Comparisonwith optimaldesign
CombinationtherapiesPartial ordering ofdoses
Models
Simulations
Using information ongrades
Illustration of the 2-group CRM shift model
Log-likelihood for power model is:
Lj(a)∆ =∑6
i=1 n1i(1)a log(αi) + n1i(0) log(1 − αai )
+∑6
i=1 n2i(1)a log(αi−∆) + n2i(0) log(1 − αai−∆) (3)
where ∆=0,1 or 2.
n1i(1) = # DLT from G1 patients on dose in1i(0) = # non-DLT from G1 patients on dose in2i(1) = # DLT from G2 patients on dose in2i(0) = # non-DLT from G2 patients on dose i
MSKCC, New York 2009 7 / 25
Bayesianmodel choice
J. O’Quigley
OverviewContinualreassessmentmethod
Subjectheterogeneity
2 group model
SimulationsImpact of prior
Comparisonwith optimaldesign
CombinationtherapiesPartial ordering ofdoses
Models
Simulations
Using information ongrades
Sample of trial history from the two-group CRMshift design
0 5 10 15 20 25 30 35
12
34
56
patient number
Dos
e le
vel
G1 non−toxicityG2 non−toxicity
G1 toxicityG2 toxicity
MSKCC, New York 2009 8 / 25
Bayesianmodel choice
J. O’Quigley
OverviewContinualreassessmentmethod
Subjectheterogeneity
2 group model
SimulationsImpact of prior
Comparisonwith optimaldesign
CombinationtherapiesPartial ordering ofdoses
Models
Simulations
Using information ongrades
Four schemes
1 Two-sample CRM shift design
2 Two-group two-parameter CRM design
3 Two separate studies design
4 One single study design, ignoring the group difference
MSKCC, New York 2009 9 / 25
Bayesianmodel choice
J. O’Quigley
OverviewContinualreassessmentmethod
Subjectheterogeneity
2 group model
SimulationsImpact of prior
Comparisonwith optimaldesign
CombinationtherapiesPartial ordering ofdoses
Models
Simulations
Using information ongrades
Target rate =0.20∆ = 0, 1, or 2
Table: Scenarios of probability of toxicity
Scenario Gr.i Probability of toxicity Ri(dk )
A 1 .07 .23 .31 .35 .45 .572 .07 .23 .31 .35 .45 .57
B 1 .08 .20 .35 .50 .70 .802 .01 .05 .18 .40 .55 .70
C 1 .02 .19 .31 .45 .51 .632 .03 .05 .11 .21 .39 .50
MSKCC, New York 2009 10 / 25
Bayesianmodel choice
J. O’Quigley
OverviewContinualreassessmentmethod
Subjectheterogeneity
2 group model
SimulationsImpact of prior
Comparisonwith optimaldesign
CombinationtherapiesPartial ordering ofdoses
Models
Simulations
Using information ongrades
Table: MTD recommendation and in-trial allocation for scenario A
Group 1 Group 2d1 d2 d3 d4 d5 d6 d1 d2 d3 d4 d5
Ri (dk ) .07 .23 .31 .35 .45 .57 .07 .23 .31 .35 .45
Proportion of MTDScheme I .27 .49 .19 .04 .00 .00 .12 .45 .28 .13 .02Scheme II .27 .47 .19 .05 .00 .00 .09 .40 .29 .16 .04Scheme III .22 .38 .23 .11 .04 .01 .22 .39 .22 .12 .04Scheme IV .17 .51 .23 .09 .01 .00 .17 .51 .23 .09 .01
Proportion of PatientsScheme I .35 .38 .19 .06 .02 .00 .19 .33 .25 .15 .06Scheme II .35 .33 .19 .07 .03 .02 .17 .29 .26 .15 .08Scheme III .31 .29 .20 .11 .06 .03 .31 .29 .20 .11 .06Scheme IV .25 .37 .22 .11 .04 .01 .25 .37 .22 .11 .04
MSKCC, New York 2009 11 / 25
Bayesianmodel choice
J. O’Quigley
OverviewContinualreassessmentmethod
Subjectheterogeneity
2 group model
SimulationsImpact of prior
Comparisonwith optimaldesign
CombinationtherapiesPartial ordering ofdoses
Models
Simulations
Using information ongrades
Table: Recommendation and in-trial allocation for scenario B
Group 1 Group 2d1 d2 d3 d4 d5 d6 d1 d2 d3 d4 d5
Ri (dk ) .08 .20 .35 .50 .70 .80 .01 .05 .18 .40 .55
Proportion of MTDScheme I .18 .54 .27 .01 .00 .00 .00 .19 .61 .19 .01Scheme II .23 .49 .26 .01 .00 .00 .00 .12 .63 .21 .03Scheme III .22 .47 .26 .04 .00 .00 .00 .15 .62 .21 .02Scheme IV .02 .43 .53 .03 .00 .00 .02 .43 .53 .03 .00
Proportion of PatientsScheme I .24 .40 .29 .06 .01 .00 .07 .22 .43 .22 .05Scheme III .32 .35 .25 .05 .02 .01 .06 .17 .44 .23 .08Scheme II .30 .33 .24 .09 .03 .01 .10 .20 .41 .20 .07Scheme IV .11 .35 .42 .09 .02 .01 .11 .35 .42 .09 .02
MSKCC, New York 2009 12 / 25
Bayesianmodel choice
J. O’Quigley
OverviewContinualreassessmentmethod
Subjectheterogeneity
2 group model
SimulationsImpact of prior
Comparisonwith optimaldesign
CombinationtherapiesPartial ordering ofdoses
Models
Simulations
Using information ongrades
Table: Recommendation and in-trial allocation for scenario C
Group 1 Group 2d1 d2 d3 d4 d5 d6 d1 d2 d3 d4 d5
Ri (dk ) .02 .19 .31 .45 .51 .63 .03 .05 .11 .21 .39
Proportion of MTDScheme I .07 .47 .39 .07 .01 .00 .00 .07 .32 .48 .12Scheme II .11 .47 .32 .08 .02 .00 .01 .06 .32 .38 .20Scheme III .11 .46 .34 .07 .01 .00 .00 .04 .30 .43 .20Scheme IV .01 .23 .55 .19 .02 .00 .01 .23 .55 .19 .02
Proportion of PatientsScheme I .16 .38 .34 .09 .03 .00 .05 .13 .29 .35 .15Scheme II .23 .34 .28 .10 .03 .02 .06 .11 .27 .30 .18Scheme II .24 .34 .25 .10 .05 .02 .10 .14 .26 .26 .16Scheme IV .08 .24 .39 .20 .06 .02 .08 .24 .39 .20 .06
MSKCC, New York 2009 13 / 25
Bayesianmodel choice
J. O’Quigley
OverviewContinualreassessmentmethod
Subjectheterogeneity
2 group model
SimulationsImpact of prior
Comparisonwith optimaldesign
CombinationtherapiesPartial ordering ofdoses
Models
Simulations
Using information ongrades
Table: Simulation results for the shift model with prior probabilitiesof shifted doses ∆
Group 1 Group 2d1 d2 d3 d4 d5 d6 d1 d2 d3 d4 d5 d6
Ri (dk ) .08 .20 .35 .50 .70 .80 .01 .05 .18 .40 .55 .70
prior = .33, .33, .33% of MTD .18 .54 .27 .01 .00 .00 .00 .19 .61 .19 .01 .00% of Patients .24 .40 .29 .06 .01 .00 .07 .22 .43 .22 .05 .01
prior = .25, .50, .25% of MTD .10 .56 .32 .01 .00 .00 .00 .12 .66 .21 .01 .00% of Patients .22 .44 .29 .04 .01 .00 .06 .19 .48 .22 .04 .01
prior = .29, .43, .29% of MTD .11 .57 .31 .01 .00 .00 .00 .14 .65 .20 .01 .00% of Patients .21 .43 .30 .04 .01 .00 .06 .21 .45 .22 .05 .01
prior = .17, .50, .33% of MTD .12 .62 .25 .00 .00 .00 .00 .09 .68 .22 .01 .00% of Patients .24 .47 .25 .04 .01 .00 .06 .18 .47 .24 .05 .01
prior = .33, .50, .17% of MTD .09 .58 .33 .01 .00 .00 .00 .15 .65 .19 .01 .00% of Patients .20 .44 .31 .04 .01 .00 .06 .22 .46 .22 .04 .00
prior = .20, .60, .20 MSKCC, New York 2009 14 / 25
Bayesianmodel choice
J. O’Quigley
OverviewContinualreassessmentmethod
Subjectheterogeneity
2 group model
SimulationsImpact of prior
Comparisonwith optimaldesign
CombinationtherapiesPartial ordering ofdoses
Models
Simulations
Using information ongrades
Comparison with optimal design
Table: Simulation results to compare the shift model and theoptimal design for Scenario A
Group 1 Group 2dk 1 2 3 4 5 6 1 2 3 4 5Rk 0.07 0.23 0.31 0.35 0.45 0.57 0.07 0.23 0.31 0.35 0.45pk (16) 0.27 0.49 0.19 0.04 0.00 0.00 0.12 0.45 0.28 0.13 0.02qk (16) 0.23 0.50 0.14 0.10 0.03 0.00 0.23 0.50 0.14 0.10 0.03
MSKCC, New York 2009 15 / 25
Bayesianmodel choice
J. O’Quigley
OverviewContinualreassessmentmethod
Subjectheterogeneity
2 group model
SimulationsImpact of prior
Comparisonwith optimaldesign
CombinationtherapiesPartial ordering ofdoses
Models
Simulations
Using information ongrades
Comparison with optimal design
Table: Simulation results to compare the shift model and theoptimal design for scenario B
Group 1 Group 2dk 1 2 3 4 5 6 1 2 3 4 5Rk 0.08 0.20 0.35 0.50 0.70 0.80 0.01 0.05 0.18 0.40 0.55pk (16) 0.18 0.53 0.27 0.01 0.00 0.00 0.00 0.19 0.61 0.19 0.01qk (16) 0.24 0.55 0.20 0.01 0.00 0.00 0.02 0.13 0.68 0.17 0.00
MSKCC, New York 2009 16 / 25
Bayesianmodel choice
J. O’Quigley
OverviewContinualreassessmentmethod
Subjectheterogeneity
2 group model
SimulationsImpact of prior
Comparisonwith optimaldesign
CombinationtherapiesPartial ordering ofdoses
Models
Simulations
Using information ongrades
Phase I study of a combination
Table: Drug combinations used in Phase 1 trial of SamariumLexidronam and Bortezomib DLT defined by as a grade 3+neutropenia (Berenson et al. 2009)
Drug CombinationAgent d1 d2 d3 d4 d5 d6Sm (mCi/kg) 0.25 0.5 1.0 0.25 0.5 1.0Bortezomib (mg/m2) 1.0 1.0 1.0 1.3 1.3 1.3
MSKCC, New York 2009 17 / 25
Bayesianmodel choice
J. O’Quigley
OverviewContinualreassessmentmethod
Subjectheterogeneity
2 group model
SimulationsImpact of prior
Comparisonwith optimaldesign
CombinationtherapiesPartial ordering ofdoses
Models
Simulations
Using information ongrades
Partial Ordering of Doses
Consider two drugs, A and B, resulting in the treatment combinations.
Drug CombinationAgent d1 d2 d3 d4 d5 d6
A 54 67.5 75 79 84.5 89.5B 6 6 6 5 7.5 9
Level d3 may be more or less toxic than d4
There are two possible simple orders (models) consistent with the partial order.
We index the models by M where M takes value Mh under the hth possible ordering
M1: d1 → d2 → d3 → d4 → d5 → d6M2: d1 → d2 → d4 → d3 → d5 → d6
MSKCC, New York 2009 18 / 25
Bayesianmodel choice
J. O’Quigley
OverviewContinualreassessmentmethod
Subjectheterogeneity
2 group model
SimulationsImpact of prior
Comparisonwith optimaldesign
CombinationtherapiesPartial ordering ofdoses
Models
Simulations
Using information ongrades
Set of possible orders of toxicity probabilities
Ordering Simple Order
M1 R(d1) ≤ R(d2) ≤ R(d3) ≤ R(d4) ≤ R(d5) ≤ R(d6)M2 R(d1) ≤ R(d2) ≤ R(d4) ≤ R(d3) ≤ R(d5) ≤ R(d6)M3 R(d1) ≤ R(d2) ≤ R(d4) ≤ R(d5) ≤ R(d3) ≤ R(d6)M4 R(d1) ≤ R(d4) ≤ R(d2) ≤ R(d3) ≤ R(d5) ≤ R(d6)M5 R(d1) ≤ R(d4) ≤ R(d2) ≤ R(d5) ≤ R(d3) ≤ R(d6)
MSKCC, New York 2009 19 / 25
Bayesianmodel choice
J. O’Quigley
OverviewContinualreassessmentmethod
Subjectheterogeneity
2 group model
SimulationsImpact of prior
Comparisonwith optimaldesign
CombinationtherapiesPartial ordering ofdoses
Models
Simulations
Using information ongrades
Escalation rules
Drug CombinationAgent 1 2 3 4 5 6Paclitaxel 54 67.5 81 94.5 67.5 67.5Carboplatin 6 6 6 6 7.5 9
d3 // d4
d1// d2
??~~~~~~~
@@@
@@@@
d5 // d6
MSKCC, New York 2009 20 / 25
Bayesianmodel choice
J. O’Quigley
OverviewContinualreassessmentmethod
Subjectheterogeneity
2 group model
SimulationsImpact of prior
Comparisonwith optimaldesign
CombinationtherapiesPartial ordering ofdoses
Models
Simulations
Using information ongrades
Probability Model
Let dhi be the dose combination at level i under ordering h
We model true probability of toxicity at dose dhi , dhi ∈ d1, . . . , dk via
R(dhi ) = Pr(Yj = 1 | dhi ) = E(Yj | dhi ) = ψh(dhi , ah)
Using simple power model:
ψh(dhi , ah) = αahhi
where 0 < αhi < 1 and 0 < ah < ∞.
For each ordering, “model”, Mh , find ah
Choose model which maximizes likelihood.
Can put priors on particular models (orderings).
MSKCC, New York 2009 21 / 25
Bayesianmodel choice
J. O’Quigley
OverviewContinualreassessmentmethod
Subjectheterogeneity
2 group model
SimulationsImpact of prior
Comparisonwith optimaldesign
CombinationtherapiesPartial ordering ofdoses
Models
Simulations
Using information ongrades
Inference
Likelihood under ordering Mh is
φ(Ωj | Mh, ah) =
j∏ℓ=1
Ψyℓ
h (xℓ, ah)1 − Ψh(xℓ, ah)(1−yℓ)
For each Mh, h = 1, . . . ,H, likelihood can be maximized in order to obtain ah , for ah .
Can use informative priors on Mh or discrete uniform
f (Mh, ah | Ωj ) ∝ φ(Ωj | Mh, ah)g(ah | Mh)p (Mh)
where
C =H∑
h=1
∫u∈A
φ(Ωj | Mh, u)g(u | Mh)p (Mh)d(u)
Marginal posterior distribution of the ordering, M, by integrating out over ah
f (Mh | Ωj ) =1
Cp (Mh)
∫A
φ(Ωj | Mh, ah)g(ah | Mh)dah.
Update in the usual way given Ωj .
Can use f (Mh | Ωj ) to sample an ordering (model).
MSKCC, New York 2009 22 / 25
Bayesianmodel choice
J. O’Quigley
OverviewContinualreassessmentmethod
Subjectheterogeneity
2 group model
SimulationsImpact of prior
Comparisonwith optimaldesign
CombinationtherapiesPartial ordering ofdoses
Models
Simulations
Using information ongrades
Simulations
Dose d1 d2 d3 d4 d5 d6 n toxR(di ) 0.26 0.33 0.51 0.62 0.78 0.86 - -
Conaway et al. 0.35 0.52 0.11 0.02 0.00 0.00 21.3 8.5POCRM 0.29 0.50 0.16 0.04 0.01 0.00 22.0 8.4
CRM 0.27 0.49 0.23 0.01 0.00 0.00 22.0 7.9
R(di ) 0.12 0.21 0.34 0.50 0.66 0.79 - -Conaway et al. 0.07 0.29 0.42 0.21 0.01 0.00 25.6 9.0
POCRM 0.02 0.23 0.55 0.11 0.10 0.00 26.0 10.0CRM 0.01 0.18 0.63 0.17 0.01 0.00 25.0 7.5
R(di ) 0.04 0.07 0.20 0.33 0.55 0.70 -Conaway et al. 0.00 0.02 0.38 0.51 0.08 0.02 28.5 8.8
POCRM 0.00 0.00 0.26 0.50 0.23 0.01 29.0 10.8CRM 0.00 0.01 0.19 0.67 0.13 0.00 28.0 8.0
R(di ) 0.01 0.04 0.05 0.17 0.33 0.67 -Conaway et al. 0.00 0.00 0.06 0.25 0.64 0.05 29.0 7.8
POCRM 0.00 0.00 0.01 0.29 0.61 0.09 29.0 9.4CRM 0.00 0.00 0.00 0.18 0.76 0.06 28.0 6.3
R(di ) 0.01 0.02 0.05 0.15 0.20 0.33 -Conaway et al. 0.00 0.00 0.01 0.04 0.37 0.59 26.2 5.8
POCRM 0.00 0.00 0.00 0.20 0.12 0.68 27.0 6.4CRM 0.00 0.00 0.00 0.05 0.26 0.69 27.0 4.3
MSKCC, New York 2009 23 / 25
Bayesianmodel choice
J. O’Quigley
OverviewContinualreassessmentmethod
Subjectheterogeneity
2 group model
SimulationsImpact of prior
Comparisonwith optimaldesign
CombinationtherapiesPartial ordering ofdoses
Models
Simulations
Using information ongrades
Probability model for grades
Model R(di ), the true probability of dose-limiting toxic response for the j th patient via:
R(di ) = Pr(Yj = 3|di ) = ψ(di , a) = αai
Model the probability of a grade 2 or grade 3 response by implementing the parameter b:
Pr(Yj = 2 or Yj = 3|di ) = ξ(di , a, b) = [αai ]b
From which the probability of a grade 2 toxicity is obtained
Pr(Yj = 2|di ) = ξ(di , a, b) − ψ(di , a) = [αai ]b − αa
i
The probability of a grade 1 toxicity follows:
Pr(Yj = 1|di ) = 1 − ξ(di , a, b) = 1 − [αai ]b
MSKCC, New York 2009 24 / 25
Bayesianmodel choice
J. O’Quigley
OverviewContinualreassessmentmethod
Subjectheterogeneity
2 group model
SimulationsImpact of prior
Comparisonwith optimaldesign
CombinationtherapiesPartial ordering ofdoses
Models
Simulations
Using information ongrades
Simulation Results
Table: Compared Frequency of Final Recommendation of aStandard CRM and a Design Using Known Information onGraded Toxicities (θ=0.25, n=25)
d1 d2 d3 d4 d5 d6Rk 0.05 0.11 0.22 0.35 0.45 0.60Standard CRM 0.00 0.13 0.53 0.30 0.04 0.00CRM using grades 0.00 0.09 0.60 0.29 0.02 0.00
MSKCC, New York 2009 25 / 25